How do I identify a monologue or dialogue in a conversation?
There are a few different ways you could frame the problem...

For example, the simplest, yet probably not the most effective, would be to treat it as a supervised classification task.

In this case, you would gather data, split it into 2 classes(binary). With one dataset consisting of dialogues and the other monologues. Obviously, this framing comes with it's own set of problems. Namely, how do you prepare the dataset, or how do you deal with live examples?

Of course, there are other ways you could tackle the problem, perhaps the detection context. It all depends on what you feel is the easiest tenable solution given the variables involved in your specific instance.